The minimum value of sin^{2} theta + cos^{2} | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Let the expression be $E = \sin^{2} 	heta + \cos^{2} 	heta + \sec^{2} 	heta + \operatorname{cosec}^{2} 	heta + 	an^{2} 	heta + \cot^{2} 	he

Vedclass · Accepted Answer

$7$

Vedclass · Answer

(D) Let the expression be $E = \sin^{2} 	heta + \cos^{2} 	heta + \sec^{2} 	heta + \operatorname{cosec}^{2} 	heta + 	an^{2} 	heta + \cot^{2} 	heta$.Using the identity $\sin^{2} 	heta + \cos^{2} 	heta = 1$,we have $E = 1 + \sec^{2} 	heta + \operatorname{cosec}^{2} 	heta + 	an^{2} 	heta + \cot^{2} 	heta$.Using $\sec^{2} 	heta = 1 + 	an^{2} 	heta$ and $\operatorname{cosec}^{2} 	heta = 1 + \cot^{2} 	heta$,we get:$E = 1 + (1 + 	an^{2} 	heta) + (1 + \cot^{2} 	heta) + 	an^{2} 	heta + \cot^{2} 	heta$.$E = 3 + 2(	an^{2} 	heta + \cot^{2} 	heta)$.By the Arithmetic Mean-Geometric Mean inequality,$	an^{2} 	heta + \cot^{2} 	heta \ge 2 \sqrt{	an^{2} 	heta \cdot \cot^{2} 	heta} = 2(1) = 2$.Therefore,the minimum value of $E = 3 + 2(2) = 3 + 4 = 7$.

The minimum value of $\sin^{2} \theta + \cos^{2} \theta + \sec^{2} \theta + \operatorname{cosec}^{2} \theta + \tan^{2} \theta + \cot^{2} \theta$ is

Similar Questions

If $A + C = B,$ then $\tan A \tan B \tan C = $

The value of $\sec ^{2} \theta - \frac{\sin ^{2} \theta - 2 \sin ^{4} \theta}{2 \cos ^{4} \theta - \cos ^{2} \theta}$ is

$\tan 7 \frac{1}{2}^{\circ}$ is equal to

If $\tan A = \frac{1 - \cos B}{\sin B}$,find $\tan 2A$ in terms of $\tan B$ and show that:

The circular measure of an angle of an isosceles triangle is $\frac{5 \pi}{9}$. The circular measure of one of the other angles must be

Vedclass Test Series

Exam Paper Generator

Online Exam Module